Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Thursday, December 27, 2012

2012 Complexity Year in Review

As we turn our thoughts from Turing to Erdős, a look back at the complexity year that was. Written with help from co-blogger Bill Gasarch.

The complexity result of the year goes to Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary and Ronald de Wolf for their paper Linear vs Semidefinte Extended Formualtions: Exponential Separation and Strong Lower Bounds (ArXiv version). It is easy to show that TSP can be expressed as an exponentially sized Linear Program (LP). In 1987 Swart tried to show that TSP could be solved with a poly sized LP. While his attempt was not successful it did inspire Yannakakis to look at the issue of how large an LP for TSP has to be.He showed in 1988 that any symmetric LP for TSP had to be exponential size. (Swart had used symmetric LP's).

What about assymetric LP's? This has been open UNTIL NOW! The paper above proves that any LP formulation of TSP requires an exponential sized LP. They use communication complexity and techniques that were inspired by quantum computing.

News and trends: The new Simons Institute for the Theory of Computing at Berkeley, the great exodus of Yahoo! researchers mostly to Google and Microsoft, the near death of Florida computer science (anyone want to be chair?), and the rise of the MOOCs.

The paper entitled 'Linear vs Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds' is really a seminal paper and I personally think that the paper somehow holds a key to establish tight hardness of approximation result for some combinatorial optimization problems, including the metric TSP which is undoubtedly one of the best open research problems. Prof. Buss and Prof. Williams's result on satisfiability is also great in its own right.Lets see what happens in next year's STOC, FOCS, CCC, ICALP or FSTTCS.

"any LP formulation of TSP requires an exponential sized LP" This cannot be what was proved. LP is P-complete, so a proof of the above statement (reasonably formalized) would appear to imply P neq NP. Can someone explain what was really proved?

The LP formulations for TSP in the paper are required to project exactly to the TSP polytope. This is a bit weaker than saying "any LP formulation of TSP requires an exponential sized LP". Essentially what it means is that you can't have a small polytope such that optimizing over it always gives you a solution for TSP for arbitrary cost-functions (You have to fix the polytope once and for all and it should work for every cost function).

Nevertheless, since "formulating TSP as an LP" *usually* is done by giving an LP whose associated polyhedral region projects exactly to the TSP polytope and a blog entry probably couldn't do justice to the fine print without alienating non-experts, quite often you'd see the above statement.

P-completeness is a space reduction, not a time reduction and hence LP not being able to solve TSP does not imply there doesn't exist a polynomial time algorithm to solve TSP. Hence the result does not imply P \neq NP